I'm just trying to learn induction from scratch and my book asks me to proof the generalized version of the triangle inequality.
If are arbitrary real numbers, then
My solution is as follows:
Let be the preposition that
and are true. is the normal triangle inequality, so i assume i don't need to show that it is true.
With the induction hypothesis i assume that is true.
We are trying to show that
this is true because according to the induction hypothesis. Also according to the transitive property of real numbers if a< b, then a+c < b+c.
Since is true then
Therefore it is true that
Is my proof correct ?
And is there a better way of doing this?